I've figured out probably the best way to integrate OBS and OpenLP. It works whether OpenLP is running locally on the same computer, and also if OpenLP is running on a separate computer (because it's all based on browser sources).
There's two components:
Dock the OpenLP web controller into OBS as a Custom Browser Dock.Add OpenLP lyrics to OBS as a Browser Source, by using a simple custom Stage View. No greenscreening is needed.
It only takes 5 minutes to set up, and I've got a full video below showing the steps and some tips.
https://www.youtube.com/watch?v=NTKpPbT_l80
Finally, to send PowerPoint or video to OBS, you can use the standard Screen Capture source (if OpenLP is local) or NDI (if OpenLP is on a different computer).
Let me know what you think!